Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) I , r/y

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Identify the quadrant given in the problem, which is Quadrant I. In this quadrant, both x and y coordinates are positive, so \(x > 0\) and \(y > 0\).

Recall the formula for \(r\), the distance from the origin to the point \((x, y)\): \(r = \sqrt{x^2 + y^2}\). Since \(x^2\) and \(y^2\) are always non-negative, \(r\) is always positive.

Analyze the ratio given: \(\frac{r}{y}\). Since \(r > 0\) and \(y > 0\) in Quadrant I, both numerator and denominator are positive.

Because both numerator and denominator are positive, the ratio \(\frac{r}{y}\) must be positive.

To confirm your understanding, sketch the coordinate plane, plot a point in Quadrant I, and visually verify that \(r\) and \(y\) are positive, reinforcing why the ratio is positive.

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주요 개념

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Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant I, both x and y are positive, which affects the sign of ratios involving these values.

Distance from Origin (r)

The distance r from the origin to a point (x, y) is given by r = √(x² + y²), which is always positive. This value represents the radius in polar coordinates and is crucial for understanding ratios involving r.

Sign of Ratios Involving Coordinates

The sign of a ratio like r/y depends on the signs of numerator and denominator. Since r is always positive, the sign of r/y depends solely on y's sign, which varies by quadrant, helping determine if the ratio is positive or negative.